📄️ 2.1 Basics
This section serves as a quick introduction, where we establish the notation and fundamental mathematical concepts such as sets, functions, and various propositions. We will not dedicate time to proving these propositions, as they are already well-established and proven in the realm of school-level and university entry-level mathematics.
📄️ 2.2 Groups
A group is a concept that combines a set with a particular operation. This set can be anything - numbers, functions, molecules, etc. The operation, which is a kind of rule, takes two elements from the set and combines them to form another element within the set.
📄️ 2.3 Rings
Thus far, our journey in abstract algebra has primarily focused on groups, which are algebraic structures defined by a single operation. Our logical next step would be exploring entities with two operations, typically denoted by addition ($+$) and multiplication ($\cdot$). The introduction of a second operation opens many possibilities in defining algebraic structures like rings, fields, modules, and vector spaces.
📄️ 2.4 Rings taxonomy
Rings can have different structures and properties because they use two operations: addition and multiplication. The diagram below shows different kinds of rings and how they fit together. We're going to look at and define a few of these types of rings, that will be useful in our discussions.
📄️ 2.5 Ring of polynomials
def: Polynomial
📄️ 2.6 Modules, Vector spaces, Algebras
In the forthcoming sections, we will omit the proofs of certain self-evident truths. For example, within the context of modules, it's understood that multiplying by
📄️ 2.7 Special kinds of rings
Local ring
📄️ 2.8 Field extensions
def: Subfield
📄️ 2.9 Galois theory
Galois group
📄️ 2.10 Finite fields
In this section, we will delve into the study of finite fields, which play a key role in cryptography.